# Bottom-Up Specification of Propositional Logic/Examples/Example 1

## Example of Bottom-Up Specification of Propositional Logic

The following is a WFF of propositional logic:

$\paren {\paren {p \land q} \implies \paren {\lnot \paren {q \lor r} } }$

## Proof

It is assumed that $p$, $q$ and $r$ are WFFs by $\mathbf W: \PP_0$.

By $\mathbf W: \text {OP}$:

$\paren {p \land q}$ is a WFF
$\paren {q \lor r}$ is a WFF

By $\mathbf W: \neg$:

$\paren {\lnot \paren {q \lor r} }$ is a WFF

By $\mathbf W: \text {OP}$:

$\paren {\paren {p \land q} \implies \paren {\lnot \paren {q \lor r} } }$ is a WFF

$\blacksquare$